A few months ago, I had someone working on my outdoor a/c unit, and he accidentally shorted out the control board in the furnace (it's a Goodman gas furnace GKS91155DX). He replaced the board, and the a/c did end up working, but when I tried to start the heat today, the burners would not stay on for longer than a few seconds before cutting out.

After ruling out the usual culprits (flame sensor, bad ground, etc.), I turned my attention to the control board, and after a little research I came to realize that it was the wrong board! The one in my furnace was PCBBF122, while the one listed in the service manual for my model is PCBBF112.

Does anybody know whether these boards are generally interchangeable? The terminals are mostly the same, except that the "wrong" board has 2 extra terminals -- one for "LO HEAT" and the other for "EAC-H".

I would go back to the person you had working on your AC unit a few months ago and get them to provide the proper replacement. I do presume that the person provided the replacement furnace controller at no charge due to their mistake at shorting out the original. Make them make it right.

On the other hand your A/C person has demonstrated several instances of questionable competence and maybe the better thing is to keep them as far away from your property as possible.

Do note that DIY sourced furnace controller boards can be substantially lower cost than those supplied through a contractor. Also check carefully on line at various sources of controller boards. I have found, through personal experience, that prices can vary as much as 1X to 3X from each other!!